Question

Consider the system y''(t) + 10y'(t) + 21y(t) = x''(t) − 4x'(t) + 5x(t), which is assumed to be causal. (35 points total). Find the transfer/system function, including the ROC.

Answer 1

The transfer/system function H(s) of the given linear differential equation is (s^2 - 4s + 5) / (s^2 + 10s + 21), and the Region of Convergence (ROC) for this causal system is s > -3.

Step-by-step explanation:

To find the transfer/system function H(s) of the given linear differential equation, we first take the Laplace transform of both sides of the equation, assuming zero initial conditions since the system is causal. The equation transforms into:

(s^2 + 10s + 21)Y(s) = (s^2 - 4s + 5)X(s)

The transfer function is then given by the ratio of the output to the input in the Laplace domain:

H(s) = Y(s) / X(s) = (s^2 - 4s + 5) / (s^2 + 10s + 21)

The Region of Convergence (ROC) for a causal system is to the right of the rightmost pole of the transfer function. To find the poles, we set the denominator of H(s) to zero:

s^2 + 10s + 21 = 0

Solving this quadratic equation, we find two poles at s = -3 and s = -7. Therefore, the ROC is s > -3.